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A note on Lang's conjecture for quotients of bounded domains

Sébastien Boucksom, Simone Diverio

Abstract

It was conjectured by Lang that a complex projective manifold is Kobayashi hyperbolic if and only if it is of general type together with all of its subvarieties. We verify this conjecture for projective manifolds whose universal cover carries a bounded, strictly plurisubharmonic function. This includes in particular compact free quotients of bounded domains.

A note on Lang's conjecture for quotients of bounded domains

Abstract

It was conjectured by Lang that a complex projective manifold is Kobayashi hyperbolic if and only if it is of general type together with all of its subvarieties. We verify this conjecture for projective manifolds whose universal cover carries a bounded, strictly plurisubharmonic function. This includes in particular compact free quotients of bounded domains.

Paper Structure

This paper contains 6 sections, 2 theorems, 4 equations.

Table of Contents

  1. Introduction
  2. The Bergman metric and manifolds of general type
  3. Proof of Theorems A and B
  4. Proof of Theorem A
  5. Proof of Theorem B
  6. Examples

Key Result

Lemma 2.2

Let $M$ be a complete Kähler manifold with a bounded psh function $\varphi$. If $\varphi$ is strictly psh on $M$ (resp. at some point of $M$), then the Bergman metric of $M$ is non-degenerate (resp. generically non-degenerate).

Theorems & Definitions (8)

  • Conjecture
  • Definition
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Remark 2.4